\HHT Vector Space h\\B#aaddcc
 
  \fts3\bVector Space \  \_\_

  

  1. Definition.\_\_

  
  Vector space is a set V with operations "+" and "." such this operations have the
  properties as follows. In other words, any two elements of this set can be added
  to each other, and each element of this set can be multiplied with a number.
  (More general theory discuss also multiplication with element from a Ring.)
  It is assumed the sum of two elements and result of multiplication with a 
  number belong V also.\_\_
  Term "vector" means simply "element of vector space".\_\_
  
  \b \Ar$f?source=vs_examples.txt" Examples. a\ b\\_\_

  Strictly speaking, "with operations "+" and "." " means that there are exist two
  functions a : V x V --> V and m : R x V --> V, and one writes the image a(u,v) as
  v+u and image m(r,v) as rv.\_\_

  \bProperties. \ (Axioms.)\_\_

  1. u + ( v + w ) = (u + v) + w for any of u,v,w belonging to V. Practically, this leads
     to "dropping" parenthesis in expressions such as (a+b)+((c+d)+w).\_\_

  2. u + v = v + u  Strictly speaking, for any v,u belonging to V, u+u=u+v.
     This property makes life very easy, one usually don't care anymore in wich order
     take sum of arbitrary number of vectors. \_\_ 
  
  3. There are elements z,z' in V that \_
     for any v, v + z = v and z' + v = v.\_
     Momentarily, we can conclude that z=z' which we will sometimes denote as 0.\_
     Momentarily, we can conclude that there is only one such z' wich will be 0.\_
     Momentarily, we can conclude that there is only one such z wich will be 0.\_
     So, we say that there is only one element 0 in V which is "neutral from the
     right and from the left".\_\_

  4. For every v, there is -w that v + -w = 0, and for every v there is -w'
     that -w' + v = 0.
     Momentarily, we can conclude that for every v, -w' = -w. Simply because\_\_
     -w = 0 + -w = (-w' + v) + -w = -w' + (v + -w) = -w' + 0 = -w' \_
     We will denote this element -w as -v (reverse to v.) \_
  
     One who wish to learn more about operations with 1, 3, and 4 properties can 
     go to Group Theory. For our vector space research, we will limit our scope
     with following four properties.

  5. For any number a and any vectors u,v,\_\_

     a( u + v ) = au + av\_\_

  6. For any numbers a,b and any vector v,\_\_
     (a + b) v  = av + bv\_\_

  7. For any numbers a,b and any vector v, \_\_
    
     (ab)v = a(bv)\_\_

  8. (-1)v = -v     \_
      -v is "reverse to v" as described in axiom 4. It seems, that axiom 8 "kills two
      birds by one stone": it defines "scale" between numbers and vectors, and "binds"
      reverse element in R with reverse element in V.\_\_

       
  \bSimple consequences. \\_\_

     According our notations --v is reverse to -v.   \_
     --v + -v =0; v = --v + -v + v = --v\_
     So, we proved that --v = v. One cannot "built the chain" of reverse elements;
     it "terminates on the second step." \_\_

     1v = ((-1)(-1))v = (-1)((-1)v) = (-1)(-v) = --v = v\_
     So, unit in R does "not change" vectors. What will happen if one defines axiom 8 as
     (-1)v = v + v ?\_\_
     
     0v = (1+(-1))v = 1v + -v = 0 \_
     This is really a great result. What will happen if one defines axiom 8 as
     (-1)v = v?\_\_

     if vector z=0, then for a <> 0, az + v = a(z + (1/a)v) = v. 
     this implies that for every a, az = 0.\_\_
     
     Axiom 8 is also important because it simplifies notations: "(-1)v" and -v are 
     interchangable.  Axioms 1, 7, and 8 greatly simplify notations containing
     parenthesis.\_\_

     \bDefinition. \ Subset V of vector space V' such in respect to operations "+" and "."  
     V is vector space itself called subspace of V'. \Ar$f?source=subspaces.txt" Examples \ \_\_

     \bDefinition. \ Linear combination (LC) of vectors v\[1 \ , v\[2 \ , ...., v\[m \  is
     a sum b\[m \ v\[1 \ + b\[m \ v\[2 \ + .... + b\[m \ v\[m \ where b are numbers. 
     Non-trivial linear combination (NLC) is linear combination 
     that at least one of numbers b is not 0.\_\_

     <a name="independent">
     \bDefinition. \ Finite set of vectors is linearly independent (sometimes we will say
     "independent", if for any NLC of this vectors is not 0. \_
     Momentarily we can obtain that set containing one vector is independent 
     when and only when this set is not {0}. \_\_

     \bTheorem. \ Any set of two or more vectors is linearly dependent when and only when
     one of this vectors is LC of others.\_\_
     
     

 \Ar$f?source=linear_algebra.txt"    \fs1#ff4400  GO TO PREVIOUS a\ \-\-\-
 \Ar$f?source=linear_algebra.txt"  \fs1#ff4400  GO TO TOP a\ \-\-\-
 \Ar$f?source=linear_functions.txt"           \fs1#ff4400  GO TO NEXT a\ \-\-\-

 \Ar$f?source=consistence.txt"  \fs1#884400  LOGICAL STRICTNESS a\
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 \Ar$f?source=disclaimer.txt"  \fs1#884400  DISCLAIMER a\    

  

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